Finite Group Databasebeta

← #877292

Subgroups — #877292

Order 512.

2022 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2570C212yesmaximal
2#17075C212yesmaximal
3#17074C212yesmaximal
4#15714C214yesmaximal
5#15347C214yesmaximal
6#15242C212yesmaximal
7#15221C212yesmaximal
8#15133C212yesmaximal
9#14712C211yesmaximal, characteristic
10#14712C211yesmaximal, characteristic
11#14446C212yesmaximal
12#14441C212yesmaximal
13#14005C212yesmaximal
14#13885C212yesmaximal
15C4×#389C211yesmaximal, characteristic
16#2570—216no—
17C2×#400C2214yes—
18C2×#398C2214yes—
19C2×#398C2214yes—
20C2×#389C2212yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#389C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28C2×#389C2214yes—
29C2×#389C2212yes—
30C2×#389C2212yes—
31C2×#387C2214yes—
32C2×#387C2214yes—
33C2×#384C2211yescharacteristic
34C2×#384C2211yescharacteristic
35C2×#384C2211yescharacteristic
36C2×#384C2211yescharacteristic
37C2×#382C2212yes—
38C2×#382C2212yes—
39C2×C4×C4⋊C4C2212yes—
40C2×C4×C4⋊C4C2212yes—
41C2×C4×C4⋊C4C2211yescharacteristic
42C2×C4×C4⋊C4C2211yescharacteristic
43C2×C43C2211yescharacteristic
44#1445C2214yes—
45#1445C2214yes—
46#1445C2214yes—
47#1445C2214yes—
48#1444C2214yes—
49#1444C2214yes—
50#1443C2214yes—
51#1442C2214yes—
52#1442C2214yes—
53#1441C2214yes—
54#1441C2214yes—
55#1441C2214yes—
56#1441C2214yes—
57#1441C2214yes—
58#1441C2214yes—
59#1436C2214yes—
60#1436C2212yes—
61#1436C2214yes—
62#1436C2212yes—
63C4⋊#95C2212yes—
64C4⋊#95C2212yes—
65C4⋊#95C2212yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2212yes—
70C4⋊#95C2211yescharacteristic
71C4⋊#95C2212yes—
72C4⋊#95C2211yescharacteristic
73C4⋊#95C2212yes—
74#1434C2214yes—
75C4×#95C2212yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2314yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2312yes—
80C22×C4⋊C4C2312yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C4⋊C4C2314yes—
84C22×C4⋊C4C2312yes—
85C22×C4⋊C4C2312yes—
86C22×C4⋊C4C2311yescharacteristic
87C22×C4⋊C4C2312yes—
88C22×C4⋊C4C2314yes—
89C22×C4⋊C4C2312yes—
90C22×C4⋊C4C2312yes—
91C22×C4⋊C4C2312yes—
92C22×C4⋊C4C2312yes—
93C22×C4⋊C4C2311yescharacteristic
94C22×C4⋊C4C2311yescharacteristic
95C22×C4⋊C4C2314yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2311yescharacteristic
98C22×C42C2312yes—
99C22×C42C2311yescharacteristic
100C22×C42C2311yescharacteristic
101C22×C42C2311yescharacteristic
102C22×C42C2312yes—
103C22×C42C2311yescharacteristic
104C22×C42C2311yescharacteristic
105C22×C42C2314yes—
106C22×C42C2311yescharacteristic
107C22×C42C2311yescharacteristic
108#400—216no—
109#398—216no—
110#398—48no—
111#389—28no—
112#389—28no—
113#389—28no—
114#389—24no—
115#389—24no—
116#389—28no—
117#389—28no—
118#389—48no—
119#389—44no—
120#389—44no—
121#389—44no—
122#387—216no—
123#387—216no—
124#384—24no—
125#384—42no—
126#384—42no—
127#384D418yes—
128#382—28no—
129#382—28no—
130C4×C4⋊C4—28no—
131C4×C4⋊C4—28no—
132C4×C4⋊C4—42no—
133C4×C4⋊C4Q818yes—
134C2×#95C2314yes—
135C2×#95C2314yes—
136C2×#95C2314yes—
137C2×#95C2314yes—
138C2×#95C2314yes—
139C2×#95C2314yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2312yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2312yes—
148C2×#95C2312yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2314yes—
158C2×#95C2312yes—
159C2×#95C2314yes—
160C2×#95C2312yes—
161C43—42no—
162C23×C4C2412yes—
163C23×C4C2414yes—
164C23×C4C2414yes—
165C23×C4C2414yes—
166C23×C4C2412yes—
167C23×C4C2414yes—
168C23×C4C2412yes—
169C23×C4C2412yes—
170C23×C4C2411yescharacteristic
171C23×C4C2411yescharacteristic
172C23×C4C2411yescharacteristic
173C23×C4C2411yescharacteristic
174C23×C4C2411yescharacteristic
175C23×C4C2411yescharacteristic
176C23×C4C2411yescharacteristic
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—28no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—28no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—24no—
202C2×C4⋊C4—28no—
203C2×C4⋊C4—28no—
204C2×C4⋊C4—24no—
205C2×C4⋊C4—24no—
206C2×C4⋊C4—24no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—42no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4—42no—
219C2×C4⋊C4—41no—
220C2×C4⋊C4—41no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—42no—
224C2×C4⋊C4—42no—
225C2×C4⋊C4—42no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—44no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—44no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4C2×D418yes—
239C2×C4⋊C4C2×D418yes—
240C2×C4⋊C4C2×D418yes—
241C2×C4⋊C4C2×Q814yes—
242C2×C4⋊C4C2×Q814yes—
243C2×C42—24no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—24no—
251C2×C42—24no—
252C2×C42—24no—
253C2×C42—22no—
254C2×C42—22no—
255C2×C42—22no—
256C2×C42—22no—
257C2×C42—24no—
258C2×C42—28no—
259C2×C42—41no—
260C2×C42—41no—
261C2×C42—41no—
262C2×C42—41no—
263C2×C42—41no—
264C2×C42—41no—
265C2×C42—44no—
266C2×C42—44no—
267C2×C42C2×D414yes—
268C2×C42C2×Q814yes—
269C2×C42C2×Q818yes—
270C2×C42#4114yes—
271C2×C42#4114yes—
272C2×C42#4114yes—
273#95—24no—
274#95—24no—
275#95—28no—
276#95—28no—
277#95—28no—
278#95—28no—
279#95—28no—
280#95—28no—
281#95—28no—
282#95—28no—
283#95—28no—
284#95—24no—
285#95—24no—
286#95—28no—
287#95—24no—
288#95—24no—
289#95—44no—
290#95—44no—
291#95—42no—
292#95—44no—
293#95—44no—
294#95—44no—
295#95—44no—
296#95—44no—
297#95—44no—
298#95—44no—
299#95C2×Q818yes—
300C24C2511yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—24no—
326C22×C4—24no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—24no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—24no—
340C22×C4—24no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—24no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—22no—
348C22×C4—24no—
349C22×C4—24no—
350C22×C4—22no—
351C22×C4—24no—
352C22×C4—22no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—24no—
356C22×C4—22no—
357C22×C4—22no—
358C22×C4—24no—
359C22×C4—24no—
360C22×C4—24no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—22no—
364C22×C4—24no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—24no—
377C22×C4—24no—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4C22×D418yes—
381C22×C4C22×D412yes—
382C22×C4C22×D412yes—
383C22×C4C22×D412yes—
384C22×C4C22×Q812yes—
385C22×C4C22×Q814yes—
386C22×C4C22×Q814yes—
387C22×C4C22×Q812yes—
388C22×C4C22×Q812yes—
389C22×C4C2×#4112yes—
390C22×C4C2×#4112yes—
391C22×C4C2×#4112yes—
392C22×C4C2×#4112yes—
393C22×C4C2×#4112yes—
394C22×C4C2×#4112yes—
395C22×C4C2×#4112yes—
396C22×C4C2×#4112yes—
397C22×C4C2×#4112yes—
398C22×C4H(5,𝔽2)12yes—
399C22×C4H(5,𝔽2)12yes—
400C22×C4H(5,𝔽2)12yes—
401C22×C42−1+412yes—
402C22×C42−1+412yes—
403C22×C42−1+412yes—
404C22×C42−1+412yes—
405C22×C42−1+412yes—
406C4⋊C4—28no—
407C4⋊C4—216no—
408C4⋊C4—216no—
409C4⋊C4—48no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—48no—
414C4⋊C4—48no—
415C4⋊C4—48no—
416C4⋊C4—48no—
417C4⋊C4—48no—
418C4⋊C4—48no—
419C4⋊C4—416no—
420C4⋊C4—48no—
421C4⋊C4—416no—
422C4⋊C4—84no—
423C4⋊C4—84no—
424C4⋊C4—84no—
425C4⋊C4—84no—
426C4⋊C4—88no—
427C4⋊C4—88no—
428C42—28no—
429C42—28no—
430C42—28no—
431C42—216no—
432C42—44no—
433C42—44no—
434C42—44no—
435C42—44no—
436C42—48no—
437C42—88no—
438C23C23×D411yescharacteristic
439C23C23×Q811yescharacteristic
440C23C22×#4111yescharacteristic
441C23C22×#4111yescharacteristic
442C23C22×#4111yescharacteristic
443C23C2×H(5,𝔽2)11yescharacteristic
444C23C2×H(5,𝔽2)11yescharacteristic
445C23C2×H(5,𝔽2)11yescharacteristic
446C23C2×2−1+411yescharacteristic
447C23C2×2−1+411yescharacteristic
448C23C2×2−1+411yescharacteristic
449C23C2×2−1+411yescharacteristic
450C23C2×2−1+411yescharacteristic
451C23#58511yescharacteristic
452C23#58511yescharacteristic
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—22no—
478C2×C4—22no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—24no—
486C2×C4—28no—
487C2×C4—28no—
488C2×C4—24no—
489C2×C4—24no—
490C2×C4—28no—
491C2×C4—24no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—24no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—44no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—44no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—44no—
520C2×C4—44no—
521C2×C4—44no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—44no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—44no—
529C2×C4—44no—
530C2×C4—44no—
531C2×C4—44no—
532C2×C4—44no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—42no—
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4#53314yes—
547C2×C4#53714yes—
548C2×C4#53914yes—
549C2×C4#54414yes—
550C2×C4#54714yes—
551C2×C4#54714yes—
552C2×C4D4×Q814yes—
553C2×C4#55014yes—
554C2×C4#55614yes—
555C2×C4#55714yes—
556C2×C4#55914yes—
557C2×C4#56414yes—
558C22C2×#53311yescharacteristic
559C22C2×#53711yescharacteristic
560C22C2×#53911yescharacteristic
561C22C2×#54411yescharacteristic
562C22#345811yescharacteristic
563C22C2×#54711yescharacteristic
564C22C2×#54711yescharacteristic
565C22C2×D4×Q811yescharacteristic
566C22C2×#55011yescharacteristic
567C22C2×#55611yescharacteristic
568C22C2×#55711yescharacteristic
569C22C2×#55911yescharacteristic
570C22#349611yescharacteristic
571C22#349711yescharacteristic
572C22#350311yescharacteristic
573C22#350311yescharacteristic
574C22#350511yescharacteristic
575C22#350511yescharacteristic
576C22#350611yescharacteristic
577C22#350711yescharacteristic
578C22#350711yescharacteristic
579C22#350711yescharacteristic
580C22#350911yescharacteristic
581C22#350911yescharacteristic
582C22#351111yescharacteristic
583C22#351111yescharacteristic
584C22C2×#56411yescharacteristic
585C22#353711yescharacteristic
586C22#353811yescharacteristic
587C22#355011yescharacteristic
588C22#355011yescharacteristic
589C22#355911yescharacteristic
590C22#355911yescharacteristic
591C22#356311yescharacteristic
592C22#356311yescharacteristic
593C4—24no—
594C4—24no—
595C4—24no—
596C4—24no—
597C4—24no—
598C4—44no—
599C4—48no—
600C4—42no—
601C4—44no—
602C4—82no—
603C4—84no—
604C4—82no—
605C4—84no—
606C4—84no—
607C4#272618yes—
608C2#3510411yescharacteristic
609C2#3613411yescharacteristic
610C2#3659811yescharacteristic
611C2#3730811yescharacteristic
612C2#3936511yescharacteristic
613C2#3948711yescharacteristic
614C2#3951211yescharacteristic
615C2#3974211yescharacteristic
616C2#3974911yescharacteristic
617C2#3976711yescharacteristic
618C2#4615711yescharacteristic
619C2#4615711yescharacteristic
620C2C2×#272611yescharacteristic
621C2#5982111yescharacteristic
622C2#5982711yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, prime radical, cosocleC24C25
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#25702
#170752
#170742
#157142
#153472
#152422
#152212
#151332
#147122
#144462
#144412
#140052
#138852
C4×#3892